We compute the Floquet Hamiltonian $H_F$ for weakly interacting fermions subjected to a continuous periodic drive using a Floquet perturbation theory (FPT) with the interaction amplitude being the perturbation parameter. This allows us to address the dynamics of the system at intermediate drive frequencies $\hslash {\omega }_D\ge V_0\ll {\mathcal{J}}_0$, where ${\mathcal{J}}_0$ is the amplitude of the kinetic term, ${\omega }_D$ is the drive frequency, and $V_0$ is the typical interaction strength between the fermions. We compute, for random initial states, the fidelity $F$ between wave functions after a drive cycle obtained using $H_F$ and that obtained using exact diagonalization (ED). We find that FPT yields a substantially larger value of $F$ compared to its Magnus counterpart for $V_0\le \hslash {\omega }_D$ and $V_0\ll {\mathcal{J}}_0$. We use the $H_F$ obtained to study the nature of the steady state of an weakly interacting fermion chain; we find a wide range of ${\omega }_D$ which leads to subthermal or superthermal steady states for finite chains. The driven fermionic chain displays perfect dynamical localization for $V_0=0$; we address the fate of this dynamical localization in the steady state of a finite interacting chain and show that there is a crossover between localized and delocalized steady states. We discuss the implication of our results for thermodynamically large chains and chart out experiments which can test our theory.

中文翻译：

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周期性驱动的弱相互作用费米子的浮球摄动理论

我们计算Floquet哈密顿量 $H_F$使用Floquet摄动理论（FPT），以连续振幅为驱动力的弱相互作用的费米子，其相互作用振幅为摄动参数。这使我们能够解决中等驱动频率下的系统动态问题$\hslash {\omega }_d\ge V_0\ll {\mathcal{Ĵ}}_0$，在哪里 ${\mathcal{Ĵ}}_0$ 是动力学项的振幅， ${\omega }_d$ 是驱动频率，并且 $V_0$是费米子之间典型的相互作用强度。对于随机初始状态，我们计算保真度$F$ 使用以下方法获得的驱动周期后的波动函数之间 $H_F$以及使用精确对角化（ED）获得的结果。我们发现FPT产生的值要大得多$F$ 与马格努斯同行相比 $V_0\le \hslash {\omega }_d$ 和 $V_0\ll {\mathcal{Ĵ}}_0$。我们使用$H_F$获得以研究弱相互作用的费米子链的稳态性质；我们发现各种各样的${\omega }_d$这会导致有限链的亚热或超热稳态。驱动的铁离子链显示出完美的动力学定位$V_0=0$; 我们解决了有限相互作用链的稳态中这种动态局部化的命运，并表明局部化和离域化稳态之间存在交叉。我们讨论了我们的结果对热力学大链的影响，并提出了可以验证我们理论的实验。